Theorem 0nnq 10934

Description: The empty set is not a positive fraction. (Contributed by NM, 24-Aug-1995.) (Revised by Mario Carneiro, 27-Apr-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
0nnq	⊢ ¬ ∅ ∈ Q

Proof of Theorem 0nnq

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelxp 5689	. 2 ⊢ ¬ ∅ ∈ (N × N)
2		df-nq 10922	. . . 4 ⊢ Q = {𝑦 ∈ (N × N) ∣ ∀𝑥 ∈ (N × N)(𝑦 ~Q 𝑥 → ¬ (2nd ‘𝑥) <N (2nd ‘𝑦))}
3	2	ssrab3 4030	. . 3 ⊢ Q ⊆ (N × N)
4	3	sseli 3927	. 2 ⊢ (∅ ∈ Q → ∅ ∈ (N × N))
5	1, 4	mto 200	1 ⊢ ¬ ∅ ∈ Q

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2145  ∀wral 3076  ∅c0 4279   class class class wbr 5103   × cxp 5653  ‘cfv 6533  2^nd c2nd 7986  Ncnpi 10854   <_N clti 10857   ~_Q ceq 10861  Qcnq 10862

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5168  df-xp 5661  df-nq 10922

This theorem is used by:  adderpq  10966  mulerpq  10967  addassnq  10968  mulassnq  10969  distrnq  10971  recmulnq  10974  recclnq  10976  ltanq  10981  ltmnq  10982  ltexnq  10985  nsmallnq  10987  ltbtwnnq  10988  ltrnq  10989  prlem934  11043  ltaddpr  11044  ltexprlem2  11047  ltexprlem3  11048  ltexprlem4  11049  ltexprlem6  11051  ltexprlem7  11052  prlem936  11057  reclem2pr  11058